Integrand size = 12, antiderivative size = 58 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6-\frac {\sqrt {1-a x^2}}{6 a^3 \sqrt {\frac {1}{1+a x^2}}} \]
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Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 30, 265, 267} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=-\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{6 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rule 30
Rule 265
Rule 267
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6+\frac {\int x^3 \, dx}{3 a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^3}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{3 a} \\ & = \frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^3}{\sqrt {1-a^2 x^4}} \, dx}{3 a} \\ & = \frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{6 a^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^4}{4 a}+\frac {\left (-1+a x^2\right ) \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )^2}{6 a^3} \]
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Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (x^{4} a^{2}-1\right )}{6 a^{2}}+\frac {x^{4}}{4 a}\) | \(60\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {3 \, a x^{4} + 2 \, {\left (a^{2} x^{6} - x^{2}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{12 \, a^{2}} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {\int x^{3}\, dx + \int a x^{5} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {x^{4}}{4 \, a} + \frac {{\left (a^{2} x^{4} - 1\right )} \sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}}{6 \, a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (69) = 138\).
Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.28 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\frac {{\left (\sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (\frac {2 \, {\left (a^{2} x^{2} + a\right )}}{a^{4}} - \frac {7}{a^{3}}\right )} + \frac {9}{a^{2}}\right )} + \frac {6 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a}\right )} a - \frac {3 \, {\left (2 \, a^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) - \sqrt {a^{2} x^{2} + a} {\left (a^{2} x^{2} - 2 \, a\right )} \sqrt {-a^{2} x^{2} + a}\right )}}{a^{2}} + \frac {3 \, {\left ({\left (a^{2} x^{2} + a\right )}^{2} - 2 \, {\left (a^{2} x^{2} + a\right )} a\right )}}{a^{2}}}{12 \, a^{3}} \]
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Time = 5.47 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx=\sqrt {\frac {1}{a\,x^2}-1}\,\left (\frac {x^6\,\sqrt {\frac {1}{a\,x^2}+1}}{6}-\frac {x^2\,\sqrt {\frac {1}{a\,x^2}+1}}{6\,a^2}\right )+\frac {x^4}{4\,a} \]
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